Download A Sparse Solver for Transport Problems running on GPUs

Using Today’s Fastest Chips to Design the Chips of
Tomorrow
Mauro Calderara, Sascha Brück, Mathieu Luisier
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Overview
 What we want to do
 How we do it
Mauro Calderara
| Apr 08 2016 |
2
Overview
 What we want to do → Quantum Transport: electrons and structures
 How we do it → How GPUs saved the day
Mauro Calderara
| Apr 08 2016 |
3
Probably you’re familiar with this
Mauro Calderara
| Apr 08 2016 |
4
Zooming in
Mauro Calderara
| Apr 08 2016 |
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The future?
(link to video: http://iis.ee.ethz.ch/~mauro/movie_SC15.avi)
Mauro Calderara
| Apr 08 2016 |
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From a somewhat more abstract POV
Device
Mauro Calderara
| Apr 08 2016 |
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From a somewhat more abstract POV
e
?
Device
Mauro Calderara
| Apr 08 2016 |
7
From a somewhat more abstract POV
e
?
e
Device
Mauro Calderara
| Apr 08 2016 |
7
From a somewhat more abstract POV
?
e
e
e
Device
Mauro Calderara
| Apr 08 2016 |
7
From a somewhat more abstract POV
?
e
e
e
Device
e
e
e
Mauro Calderara
| Apr 08 2016 |
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This is what we’re ultimately interested in!
 How do electrons behave w.r.t the
device?
Device
Mauro Calderara
| Apr 08 2016 |
8
This is what we’re ultimately interested in!
 How do electrons behave w.r.t the
device?
Device
 Change in parameters → change in
behavior?
Mauro Calderara
| Apr 08 2016 |
8
This is what we’re ultimately interested in!
 How do electrons behave w.r.t the
device?
e
e
e
e
e
 Change in parameters → change in
behavior?
Device
e
Mauro Calderara
| Apr 08 2016 |
8
This is what we’re ultimately interested in!
 How do electrons behave w.r.t the
device?
Gate voltage
e
e
e
e
e
 Change in parameters → change in
behavior?
Device
e
Mauro Calderara
| Apr 08 2016 |
8
This is what we’re ultimately interested in!
Gate voltage
e
Material
properties
e
e
e
 Change in parameters → change in
behavior?
Device
e
 How do electrons behave w.r.t the
device?
e
Dimensions
Mauro Calderara
| Apr 08 2016 |
8
This is what we’re ultimately interested in!
Gate voltage
e
Material
properties
e
e
e
 Change in parameters → change in
behavior?
Device
e
 How do electrons behave w.r.t the
device?
e
Dimensions
 Applies not just to transistors
 Batteries
 Storage devices
 ...
Mauro Calderara
| Apr 08 2016 |
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How would we do that? The ‘‘easy’’ case:
Mauro Calderara
| Apr 08 2016 |
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How would we do that? The ‘‘easy’’ case:
→ device behaves like bulk material
Mauro Calderara
| Apr 08 2016 |
9
How would we do that? The ‘‘difficult’’ case:
Mauro Calderara
| Apr 08 2016 |
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How would we do that? The ‘‘difficult’’ case:
→ device behaves like atomic structure
Mauro Calderara
| Apr 08 2016 |
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The cost of going small
Why is this ‘‘easy’’ ...
... and this ‘‘difficult’’?
Mauro Calderara
| Apr 08 2016 |
11
The cost of going small
Can assume is ‘‘infinite’’ and
use semi empirical model.
Very finite! Need to do
it from first principles.
Mauro Calderara
| Apr 08 2016 |
12
The cost of going small
Can assume is ‘‘infinite’’ and
use semi empirical model.
Very finite! Need to do
it from first principles.
Mauro Calderara
| Apr 08 2016 |
12
The cost of going small
Can assume is ‘‘infinite’’ and
use semi empirical model.
Very finite! Need to do
it from first principles.
Mauro Calderara
| Apr 08 2016 |
12
The cost of going small
Can assume is ‘‘infinite’’ and
use semi empirical model.
Very finite! Need to do
it from first principles.
Mauro Calderara
| Apr 08 2016 |
12
The cost of going small
Can assume is ‘‘infinite’’ and
use semi empirical model.
Very finite! Need to do
it from first principles.
Mauro Calderara
| Apr 08 2016 |
12
The cost of going small
Can assume is ‘‘infinite’’ and
use semi empirical model.
Very finite! Need to do
it from first principles.
Mauro Calderara
| Apr 08 2016 |
12
Can assume is ‘‘infinite’’ and
use semi empirical model.
runtime
runtime
The cost of going small
Very finite! Need to do
it from first principles.
Mauro Calderara
| Apr 08 2016 |
12
Semi-empirical → O(Hours)
runtime
runtime
The cost of going small
First principles → O(Months)
Mauro Calderara
| Apr 08 2016 |
13
Semi-empirical → O(Hours)
runtime
runtime
The cost of going small
First principles → O(Months)
Mauro Calderara
| Apr 08 2016 |
13
Semi-empirical → O(Hours)
runtime
runtime
The cost of going small
First principles → O(Months)
Mauro Calderara
| Apr 08 2016 |
13
Overview
 What we want to do → Quantum Transport: electrons and structures
 How we do it → How GPUs saved the day
Mauro Calderara
| Apr 08 2016 |
14
runtime
Where does all that time go?
~ 40x
Mauro Calderara
| Apr 08 2016 |
15
runtime
Where does all that time go?
~ 40x
Solve an eigenvalue
problem (not discussed
here).
Mauro Calderara
| Apr 08 2016 |
15
runtime
Where does all that time go?
~ 40x
Invert the matrix from
before (selectively!) using
a recursive algorithm.
Solve an eigenvalue
problem (not discussed
here).
Mauro Calderara
| Apr 08 2016 |
15
Avoiding the inversion, use a sparse solver instead
runtime
 Instead of trying to invert selectively,
solve system using generic sparse
solver package
~ 40x
Mauro Calderara
| Apr 08 2016 |
16
runtime
Avoiding the inversion, use a sparse solver instead
~ 40x
 Instead of trying to invert selectively,
solve system using generic sparse
solver package
 Gain: speed, parallelism, capacity for
somewhat larger systems
Mauro Calderara
| Apr 08 2016 |
16
runtime
Avoiding the inversion, use a sparse solver instead
~ 40x
 Instead of trying to invert selectively,
solve system using generic sparse
solver package
 Gain: speed, parallelism, capacity for
somewhat larger systems
 Cost: code now mem-bw bound
And: not such a good fit for GPUs ... 
Mauro Calderara
| Apr 08 2016 |
16
runtime
Avoiding the inversion, use a sparse solver instead
~ 40x
 Instead of trying to invert selectively,
solve system using generic sparse
solver package
 Gain: speed, parallelism, capacity for
somewhat larger systems
 Cost: code now mem-bw bound
And: not such a good fit for GPUs ... 
Mauro Calderara
| Apr 08 2016 |
16
runtime
runtime
Tackling the eigenvalue problem
~ 200x
 We’ve been able to solve that one 
Mauro Calderara
| Apr 08 2016 |
17
 Good speedup so far
(now: O(Days), still not
quite there...)
runtime
Now what?
~ 70x overall
Mauro Calderara
| Apr 08 2016 |
18
 Good speedup so far
(now: O(Days), still not
quite there...)
runtime
Now what?
~ 70x overall
 But
Mauro Calderara
| Apr 08 2016 |
18
 Good speedup so far
(now: O(Days), still not
quite there...)
 But
runtime
Now what?
~ 70x overall
?
Mem-BW bound by sparse solver
Mauro Calderara
| Apr 08 2016 |
18
 Good speedup so far
(now: O(Days), still not
quite there...)
 But
runtime
Now what?
~ 70x overall
?
Mem-BW bound by sparse solver
Mauro Calderara
| Apr 08 2016 |
18
 Good speedup so far
(now: O(Days), still not
quite there...)
runtime
Now what?
~ 70x overall
 But
Mem-BW bound by sparse solver
Mauro Calderara
| Apr 08 2016 |
18
 Good speedup so far
(now: O(Days), still not
quite there...)
runtime
Now what?
~ 70x overall
Advisor
 But
PhD student
?
Mem-BW bound by sparse solver
Mauro Calderara
| Apr 08 2016 |
18
A Sparse Solver for Transport Problems running on GPUs
 Inverting sparse system not feasible
-1
=
Mauro Calderara
| Apr 08 2016 |
19
A Sparse Solver for Transport Problems running on GPUs
 Inverting sparse system not feasible
 In our case: also not neccessary
-1
=
Mauro Calderara
| Apr 08 2016 |
19
A Sparse Solver for Transport Problems running on GPUs
 Inverting sparse system not feasible
 In our case: also not neccessary
-1
=
 Need first and last block rows only
Mauro Calderara
| Apr 08 2016 |
19
A Sparse Solver for Transport Problems running on GPUs
 Inverting sparse system not feasible
 In our case: also not neccessary
-1
=
 Need first and last block rows only
 If we can compute this fast, we can
 interleave the solving step with the BC
computation
 obtain the full solution very efficiently
Mauro Calderara
| Apr 08 2016 |
19
Obtaining the first and last block columns of the inverse
for i = N:1
𝑋𝑖 ← (𝐴𝑖,𝑖 − 𝐴𝑖,𝑖+1 𝑋𝑖+1 ) \ 𝐴𝑖,𝑖−1
 Recursive algorithm based on the
Schwinger-Dyson equation
for i = 2:N
𝑄𝑖 ← −𝑋𝑖 𝑄𝑖−1
Mauro Calderara
| Apr 08 2016 |
20
Obtaining the first and last block columns of the inverse
for i = N:1
𝑋𝑖 ← (𝐴𝑖,𝑖 − 𝐴𝑖,𝑖+1 𝑋𝑖+1 ) \ 𝐴𝑖,𝑖−1
 Recursive algorithm based on the
Schwinger-Dyson equation
for i = 2:N
𝑄𝑖 ← −𝑋𝑖 𝑄𝑖−1
Mauro Calderara
| Apr 08 2016 |
20
Obtaining the first and last block columns of the inverse
 Recursive algorithm based on the
Schwinger-Dyson equation
for i = N:1
𝑋𝑖 ← (𝐴𝑖,𝑖 − 𝐴𝑖,𝑖+1 𝑋𝑖+1 ) \ 𝐴𝑖,𝑖−1
for i = 2:N
𝑄𝑖 ← −𝑋𝑖 𝑄𝑖−1
N-2
N-1
N
N+1
𝐴
𝑋
Mauro Calderara
| Apr 08 2016 |
20
Obtaining the first and last block columns of the inverse
 Recursive algorithm based on the
Schwinger-Dyson equation
for i = N:1
𝑋𝑖 ← (𝐴𝑖,𝑖 − 𝐴𝑖,𝑖+1 𝑋𝑖+1 ) \ 𝐴𝑖,𝑖−1
for i = 2:N
𝑄𝑖 ← −𝑋𝑖 𝑄𝑖−1
N-2
N-1
N
N+1
𝐴
𝑋
Mauro Calderara
| Apr 08 2016 |
20
Obtaining the first and last block columns of the inverse
for i = N:1
𝑋𝑖 ← (𝐴𝑖,𝑖 − 𝐴𝑖,𝑖+1 𝑋𝑖+1 ) \ 𝐴𝑖,𝑖−1
 Recursive algorithm based on the
Schwinger-Dyson equation
for i = 2:N
𝑄𝑖 ← −𝑋𝑖 𝑄𝑖−1
 xGEMM + xGESV + xGEMM
N-2
N-1
N
N+1
𝐴
𝑋
Mauro Calderara
| Apr 08 2016 |
20
Obtaining the first and last block columns of the inverse
for i = N:1
𝑋𝑖 ← (𝐴𝑖,𝑖 − 𝐴𝑖,𝑖+1 𝑋𝑖+1 ) \ 𝐴𝑖,𝑖−1
 Recursive algorithm based on the
Schwinger-Dyson equation
for i = 2:N
𝑄𝑖 ← −𝑋𝑖 𝑄𝑖−1
 xGEMM + xGESV + xGEMM
 Very fast on accelerators
N-2
N-1
N
N+1
𝐴
𝑋
Mauro Calderara
| Apr 08 2016 |
20
Obtaining the first and last block columns of the inverse
for i = N:1
𝑋𝑖 ← (𝐴𝑖,𝑖 − 𝐴𝑖,𝑖+1 𝑋𝑖+1 ) \ 𝐴𝑖,𝑖−1
 Recursive algorithm based on the
Schwinger-Dyson equation
for i = 2:N
𝑄𝑖 ← −𝑋𝑖 𝑄𝑖−1
 xGEMM + xGESV + xGEMM
 Very fast on accelerators
N-2
N-1
N
N+1
𝐴
𝑋
 Parallelizable
Mauro Calderara
| Apr 08 2016 |
20
A Sparse Solver for Transport Problems running on GPUs
 Runs on GPUs, compute bound
Mauro Calderara
| Apr 08 2016 |
21
A Sparse Solver for Transport Problems running on GPUs
 Runs on GPUs, compute bound
Performance
[log(FLOPS)]
Arithmetic Intensity
[log(FLOPS/Byte)]
Mauro Calderara
| Apr 08 2016 |
21
A Sparse Solver for Transport Problems running on GPUs
 Runs on GPUs, compute bound
Performance
[log(FLOPS)]
 Interleaves with EV computation
Arithmetic Intensity
[log(FLOPS/Byte)]
Mauro Calderara
| Apr 08 2016 |
21
A Sparse Solver for Transport Problems running on GPUs
 Runs on GPUs, compute bound
Performance
[log(FLOPS)]
 Interleaves with EV computation
 Memory efficient
Arithmetic Intensity
[log(FLOPS/Byte)]
Mauro Calderara
| Apr 08 2016 |
21
A Sparse Solver for Transport Problems running on GPUs
 Runs on GPUs, compute bound
Performance
[log(FLOPS)]
 Interleaves with EV computation
 Memory efficient
 Much faster than sparse solvers
Arithmetic Intensity
[log(FLOPS/Byte)]
Mauro Calderara
| Apr 08 2016 |
21
A Sparse Solver for Transport Problems running on GPUs
 Runs on GPUs, compute bound
Performance
[log(FLOPS)]
 Interleaves with EV computation
 Memory efficient
 Much faster than sparse solvers
Arithmetic Intensity
[log(FLOPS/Byte)]
 Whole simulation: O(Hours)
Mauro Calderara
| Apr 08 2016 |
21
A Sparse Solver for Transport Problems running on GPUs
 Runs on GPUs, compute bound
Performance
[log(FLOPS)]
 Interleaves with EV computation
 Memory efficient
~ 10x
/ 80x
 Much faster than sparse solvers
Arithmetic Intensity
[log(FLOPS/Byte)]
 Whole simulation: O(Hours)
Mauro Calderara
| Apr 08 2016 |
21
Summary
Mauro Calderara
| Apr 08 2016 |
22
Summary
 Transforming a sparse problem to a dense one can be a good thing
Mauro Calderara
| Apr 08 2016 |
22
Summary
 Transforming a sparse problem to a dense one can be a good thing
 Large speedup over state of the art (15x - 150x)
Mauro Calderara
| Apr 08 2016 |
22
Summary
 Transforming a sparse problem to a dense one can be a good thing
 Large speedup over state of the art (15x - 150x)
 Significant increase in capacity (100’000 atoms → 10x - 100x)
Mauro Calderara
| Apr 08 2016 |
22
Summary
 Transforming a sparse problem to a dense one can be a good thing
 Large speedup over state of the art (15x - 150x)
 Significant increase in capacity (100’000 atoms → 10x - 100x)
 Uses hybrid ressources very efficiently (15 PF sustained)
Mauro Calderara
| Apr 08 2016 |
22
Summary
 Transforming a sparse problem to a dense one can be a good thing
 Large speedup over state of the art (15x - 150x)
 Significant increase in capacity (100’000 atoms → 10x - 100x)
 Uses hybrid ressources very efficiently (15 PF sustained)
 Made ballistic ab-initio QT simulations for realistic structures a reality
Mauro Calderara
| Apr 08 2016 |
22
(link to video: http://iis.ee.ethz.ch/~mauro/movie_Ag_Switch.avi)
Mauro Calderara
| Apr 08 2016 |
23
Questions?
Mauro Calderara
| Apr 08 2016 |
24